250 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF of plates may be supposed to increase and their thickness to decrease indefinitely, while at the same time the steps by which the doubly refracting nature of the plates alters from one to another become separately insensible, the theorem will be true if the whole train, or part of it, consist of a substance of which the doubly refracting nature alters continuously, as for example a piece of strained or unannealed glass. If, however, the train contain a member which performs a partial analysis of the light, as for example a plate of smoky quartz, or a plate of glass inclined to the incident light at a considerable angle, it will no longer be true that two pencils going in oppositely polarized will come out oppositely polarized. 13. THEOREM. If two equivalent groups of polarized streams be resolved in any manner, which is the same for both, into two oppositely polarized groups, and these be recombined after the phase of one of the components has been retarded by a given quantity relatively to that of the other, the two groups of resultant streams will be equivalent. Let the groups of resultants be each resolved in any manner into two oppositely polarized streams, and call these 0, E. By Art. 11, if 0', E' be the streams which furnish 0, E respectively, 0', Er are oppositely polarized. Now by Art. 3, the intensities of 0, E are the same respectively as those of 0', E'\ but these are the same for the one group as for the other, by the definition of equivalence. Therefore the intensities of 0, E are the same for the one group as for the other; but 0, E are any two oppositely polarized components of the resultant groups; therefore these groups are equivalent. Hence, if two equivalent groups be transmitted through a crystalline plate, the emergent groups will be equivalent; and by the same reasoning as in Art. 12 the theorem may be extended to an optical train consisting of any number of crystalline plates, pieces of unannealed glass, &c. 14. THEOREM. If two equivalent groups be resolved in any manner, the same for both, into two polarized streams, the intensities of the components of the one group will be respectively equal to the intensities of the components of the other group. The proof of this theorem is very easy. It is sufficient to treat the general expressions (13) exactly as in Art. 9, only that